Remark 6  [Collapsing] Note that this collapsing operation is adaptation of a well-known notion called deformation retract in algebraic topology in a simplicial complex setting [1]}. This operation keeps the homotopy type the same, and hence the homology does not change with this reduction. In [2]}, [3]}, [4]}, this is called folding (\(\mathcal {G}\)  folds onto \(\mathcal {G}-\lbrace u\rbrace \) ) or a strong collapse. In these papers, the algorithm reduces simplicial complexes in the filtration one by one so that its associated clique complex keeps the same homotopy type. Our contribution here is to adapt this operation to the graph filtrations and define a smaller subgraph before the filtration step so that the induced simplicial complexes are homotopy equivalent. Since we prune the graph at the beginning of the process, our algorithm significantly reduces the computational costs for the induced persistence diagrams.

